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CoompmlexpWleKBxAWpprKoxBimaAtiopnspinrGorxapimhenaetEiloecntrosn-iHnoleGWraavepguhidesnien MEaglnectictrFoielnd -Hole Waveguides in Magnetic Field 917 5. Conclusion Using the basic steps of techniques of Gaussian beams summation, known for acoustic wave propagation, this method was applied for electron-holes motion described by Dirac system in magnetic field and arbitrary potential in graphene to construct the Green’s tensor semiclassical uniform approximation. This approximation was tested for a special cases of waveguide excitation by point source for electron-hole motion in magnetic and linear electric fields. The asymptotic results for the Green’s tensor computed by Gaussian beams summation were found to be in a very good agreement with data obtained by the ray asymptotic solution. The method of Gaussian beams summation is efficient for construction of WKB approximation describing electron-hole motion in magnetic field and any scalar potential including problems of electron-hole waveguide transport through resonators. It is worth to remark that the semiclassical analysis for the Dirac system considered in this chapter corresponds to the case of the first K Diract point in the first Brillouine zone. The case of the second K′ Dirac point is treated in a similar way. 6. Acknowledgments The authors would like to thank Dr A.Vagov, Prof M.Popov, Prof F.Kusmartsev for constructive discussions and valuable remarks. 7. References [1] Datta, S., 1995 Electronic transport in mesoscopic systems, Cambridge University Press, Cambridge. [2] Mello,P.A.,Kumar,N.,2004QuantumTransportinMesoscopicSyStem,OxfordUniversity Press, New York. [3] Stockmann, H. J., 2000, Quantum Chaos. An Introduction, Cambridge University Press, Cambridge. [4] V. V. Belov and S.Yu. Dobrokhotov, Sov. Math. Doclady 37, 264 (1988). [5] CastroNeto,A.H.,Guinea,F.,Peres,N.M.R.,Geim,A.K.,andNovoselov,K.S.Rev.Mod. Phys. 81, Jan-March, (2009). [6] Cheianov, V.V., Fal’ko, V.I. Phys.Rev. B. 74, 041403(R)(2006). [7] Fistul, M.V., Efetov, K.B. PRL 98, 256803 (2007). [8] Silvestrov, P.G., Efetov, K.B. PRL 98, 016802 (2007). [9] Sonin, E.B. Phys.Rev. B. 79, 195438 (2009). [10] Schwieters, C.D., Alford, J.A. and Delos, J.B. Phys.Rev.B. 54, N15, 10652 (1996). [11] Beenaker, C.W.J. Rev. Mod. Phys. 69, 731 (1997). [12] Blomquist, T. and Zozoulenko, I. V. Phys.Rev.B. 61, N3, 1724 (2000). [13] Jalabert, R. A., Baranger, H. U., Stone, A. D. Phys.Rev.Lett. 65, N19, 2442 (1990). [14] Carmier, P., Ullmo, D. Phys.Rev. B. 77, N24, 245413 (2008). [15] Maslov, V. P. and M.V.Fedoriuk, 1981, Semiclassical approximation in quantum mechanics, Reidel, Dordrecht. [16] Popov, M.M., Wave Motion 61, N3, 1724 (1981). [17] Popov,M.M.,2002,MethodGaussianbeams,PetroleumBrazil. [18] V. V. Zalipaev, F. V. Kusmartsev, and M. M. Popov, J. Phys. A: Math. Theor. 41, 065101 (2008).PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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